To a Java-based Trochoid animation
These things are trochoids of a sort - the trail left by a point on a wheel as it rolls around another wheel.
Normally trochoids are divided into two categories: Epitrochoids and hypotrochoids.
Epitrochoids are the trail from a wheel rolling around the outside of a fixed circle.
Hypotrochoids are those formed from a wheel rolling around the inside of a fixed circle.
These curves are from a more general class, with the circle being rolled around also turning.
The general equations for these are:
x = size1*sin (q) + size2*sin(n*q-f)
y = size1*cos (q) + size2*cos(n*q+f))
where f depends on time.
By the way, if you watch carefully you should be able to see that all the rings in each of these animations are really the same.
To my mathematical stills
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